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Quadratic Twists of Elliptic Curves
Patricia L. Quattrini
CONTENTS 1. Introduction
2. Restatement and Procedure 3. An Example
4. Experiment and Observations References
2000 AMS Subject Classification:Primary 11F33; Secondary 11Y70 Keywords: Elliptic curves, Tate-Shafarevich groups,
modular forms
The aim of this paper is to analyze the distribution of analytic (and signed) square roots ofX values on imaginary quadratic twists of elliptic curves.
Given an elliptic curveEof rank zero and prime conductor N, there is a weight-32 modular formgassociated with it such that the d-coefficient of g is related to the value ats = 1of theL-series of the(−d)-quadratic twist of the elliptic curveE.
Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integersdthe order ofXof the(−d)-quadratic twist ofEand analyze their distribution.
1. INTRODUCTION
LetEbe an elliptic curve of prime conductorNand rank zero, and let
L(E, s) =
n≥1
ann−s
be the L-function of E. By recent work of Wiles and others it is known that
f = ∞ m=1
amqm, q=e2πiτ,
is a modular form of weight 2, level N, and L(E, s) = L(f, s). Moreover,f is a cusp form and an eigenfunction for all the Hecke operators Tn acting on M2(Γ0(N)). If we normalize to havea1= 1, then the Fourier coefficients ai are the eigenvalues.
Forda positive integer such that −d≡0,1 (mod 4), we denote by−dthe quadratic character of (Z/dZ)∗ de- termined by−d(p) =
−d p
for primesp. Then f ⊗−d=
m≥1
am−d(m)qm,
the twist off, is the weight-2, level-dividing-N d2 modu- lar form of the (−d)-quadratic twist of the curveE, which we will denote byEd; andL(Ed, s) =L(f⊗−d, s).
c A K Peters, Ltd.
1058-6458/2006$0.50 per page Experimental Mathematics15:3, page 355
A special case of Waldspurger’s formula appears in [Gross 87] that relates the product L(f,1)L(f ⊗−d,1) to the squaredd-coefficient of a weight-32 modular form g under Shimura correspondence tof.
Moreover, Gross states a procedure for calculating the weight-32 form g given the weight-2, level-N mod- ular form f. This procedure comes from the connection between modular forms on Γ0(N) and the quaternion al- gebraB ramified at N and∞.
Basically, one constructs the theta series associated with certain rank-3 lattices in the quaternion algebra.
These theta series are then modular forms of weight
3
2. The weight-32 modular form g corresponding to the weight-2 modular form f will be a linear combination of these, given by the coefficients of an eigenvectorvof the Brandt matrices of level N.
We will briefly state this procedure, following the no- tation in [Gross 87]. Consider a maximal order R in the quaternion algebraB, ramified atN and∞, and let {I1, . . . , In} be a set of left-ideal classes for the orderR.
LetRi denote the right (maximal) order for the idealIi. For each right orderRi, one constructs the theta series gi of the following rank-threeZ-lattices: DefineSi0as the subgroup of elements of trace zero in the Ri-suborder Z+ 2Ri. LetNdenote the norm form, which is positive definite. Then define the theta seriesgi by
gi(τ) = 1 2
b∈Si0
qNb= 1
2+
d>0
ai(d)qd, i= 1, . . . , n.
If t is the type number, that is, the number of distinct conjugacy classes of maximal orders inB, one getstdif- ferent theta series gi (some are repeated).
These theta series are modular forms of weight 32, level 4N, and trivial character, and the coefficientsad satisfy
ad= 0 unless −d≡0 (mod 4) and −d
N
= 1. () This is the Kohnen subspace of modular forms of weight
3
2, which has dimensiont and is stable under the Hecke algebra. The theta series lie in a lattice of rankt, denoted byM∗, which consists of those forms that in addition to satisfying (), have integral coefficients, except (possibly) fora0, which lies in 12Z.
1.1 Jacquet–Langlands Correspondence
LetN be a prime andS2(Γ0(N)) the space of cusp forms of weight 2 for the congruence subgroup Γ0(N). The Hecke operators Tp act on this space, and we are inter-
ested in eigenformsf: f =
∞ n=1
anqn (a1= 1), Tp(f) =apf.
To each of these modular formsf corresponds an eigen- vector of the Brandt matrices of prime level N: there is a dimension-one space of eigenvectors v such that B(N, p)v=apv for all primesp(and for alln∈N).
Letv = (v1, . . . , vn) be (up to a constant factor) the eigenvector of the Brandt matrices of prime levelNcorre- sponding to ourf ∈S2(Γ0(N)) coming from the elliptic curveE.
Defineef = (e1, . . . , en) = (v1/w1, . . . , vn/wn), where wi is one-half the number of units of the orderRi.
Then
g= n i=1
eigi =
d>0
mdqd
is the weight-32 modular form that corresponds to the weight-2 modular formf. And this is the form involved in the Waldspurger formula, as related by Gross.
1.2 Waldspurger’s Formula
LetN be a prime number,f a cusp form of weight 2 and trivial character for Γ0(N), and f ⊗−d the twist of f by the character−dof (Z/dZ)∗determined by−d(p) = −d
p
for primesp. Forda positive integer such that−d is a fundamental discriminant and−d
N
= 1, we have
L(f,1)L(f⊗−d,1) =kN (f, f)m2d
√def, ef,
where kN = 2 if d ≡ 0 (modN) and is 1 otherwise, and (f, f) is the Peterson product. The product. , .is defined as follows: ifv = ni=1viei and u= ni=1uiei, then
v, u= n i=1
wiviui.
2. RESTATEMENT AND PROCEDURE
Now we will restate the above formula in a more conve- nient way. We can replace the product (f, f) in the for- mula above if we takeEto be a strong Weil curve and use the following result, which can be found in [Cremona 95]:
Letf(τ) be a normalized new form of weight 2 for Γ0(N).
The periods of 2πif(τ) form in this case a lattice Λf, and the modular elliptic curveEf =C/Λf is defined over Q
and has conductor N. Let ϕ: X = Γ0(N)\H∗ −→Ef be the associated modular parameterization. Then
4π2f2= deg(ϕ) Vol(Ef), where
f2=
X
|f(τ)|2du dv, τ =u+iv.
If we write the lattice Λf as generated by Ωf(1, τ), where Ωf is the real period of Ef, then
Vol(Ef) = Ω2fIm(τ),
where Im(z) denotes the imaginary part ofzand Vol(Ef) is the volume of the lattice Λf. Replacing this in Wald- spurger’s formula and rearranging in a convenient way, we have
L(f,1) Ωf
L(f⊗−d,1)
2Ωf√Im(τ) d
=kN deg(ϕ)m2d ef, ef ,
for−d
N
= 1,where deg(ϕ) is taken from Cremona’s ta- bles. The factor L(f,1)Ω
f on the left is a rational number, which we can calculate with PARI-GP. The denominator
2Ωf√Im(τ)
d is “almost” the real period of the (−d)-twist of f,f⊗−d. That is, it is a rational calculable multiple of it that depends sometimes on dmodulo 8.
2.1 BSD: The Rank-Zero Case
For an elliptic curve E of rank zero, the Birch and Swinnerton-Dyer (BSD) conjecture states that
L(E,1)
Ω =|X|
pcp
|Tor(E)|2,
where Tor(E) is the torsion group of E, and Ω is the integral overE(R) of the invariant differentialω. This is the real period ofE or twice the real period, depending on whether the polynomialp(x) definingE in y2=p(x) has negative or positive discriminant.
ForEd, the (−d)-twist of the elliptic curveE, we have L(Ef⊗−d,1) =L(Ed,1), and we substitute the equality above given by the BSD conjecture into Waldspurger’s formula. We then have
L(E,1) Ω
|Xd|
pcp,d
|Tor(Ed)|2 qd= kN2 deg(ϕ)m2d ef, ef . The order of Tor(Ed) is constant. The factorqdis the ra- tional multiple that comes from the quotient ΩdΩfIm(τ)
√d . In calculated examples,qd is the constant 2, or it equals 2 or 4 depending on the divisibility ofdmodulo 8. The
quotient ofL(E,1), the period Ω, and the product of the fudge factors
pcp,d can be calculated with PARI-GP.
This is the identity we use to calculate the order of Xd for a family of imaginary quadratic twists of the el- liptic curveE, having previously calculated the weight-32 modular formg:
g(τ) =
d>0
mdqd.
Moreover, as is known, the order of the Tate–Shafarevich group is a square. Then we can get signed square roots of Xd with the sign given by the md-coefficient of the weight-32 modular formg.
We will concentrate our attention on arithmetic as- pects of the density distribution of the square roots of |X| values obtained in the family of (−d)-quadratic twists of an elliptic curveE. There are relevant results concerning the distribution of values of L-series due to Conrey, Keating, Farmer, Rubinstein, Snaith, and De- launay that will not be mentioned here. For references see [Conrey et al. 04, Delaunay 01].
3. AN EXAMPLE
We will write explicitly all calculations for the prime con- ductorN = 17. From Cremona’s tables we see that there is one isogeny class of elliptic curves of conductorN = 17 and rank zero. We take the curve number one in the class, that is, the strong Weil curve, in order to use the above formula to calculate (f, f). This is, in the format [a1, a2, a3, a4, a6] given by
y2+a1xy+a3y=x3+a2x2+a4x+a6,
the curveE= [1,−1,1,−1,−14]; its torsion has four el- ements, and the degree of the modular parameterization ϕis 1.
With this curve we have to associate a modular form g of weight 32, which will be a linear combination of the theta series gi of the lattices Si0 constructed from right ordersRiin the quaternion algebra ramified at 17 and∞. 3.1 Calculation of the Theta Seriesgi
For our calculations we used routines from A. Pacetti’s qalgmodforms [Pacetti 01] for doing arithmetic over quaternion algebras and from G. Tornaria’s qftheta3 [Tornaria 04], both of which run under PARI-GP. We will employ the following notation: [b0, b1, b2, b3] stands forb0+b1i+b2j+b3k in the quaternion algebra.
First, we take a maximal order R in the quaternion algebra ramified at N = 17 and ∞ and calculate a set
of representativesI1, . . . , In of left ideals for the orderR.
For each of these ideals Ii we calculate the right (maxi- mal) ordersRi. We then take the trace-zero elements of the lattices Z+ 2Ri. The modular forms gi will be the theta series of these lattices. This can be done in the following way with routines from the packages above:
qsetprime(17) sets the quaternion algebra ramified at N = 17 and∞, and returns a maximal orderR in it:
R=
[12,0,12,0],[0,12,0,12],[0,0,13,13],[0,0,0,1],1 .
Thenqdef tells us which quaternion algebra we are in:
qdef= [−17,−3].
That is, the quaternion algebra ramified at N = 17 and
∞is
B={b0+b1i+b2j+b3k, bi∈Q, i2=−17, j2=−3}.
We then have to calculate a set of representatives of left ideals for the maximal orderR:
qidcl(R) = 12,0,12,0 ,
0,12,0,12 ,
0,0,13,13 , [0,0,0,1],1
,
[1,0,1,0],[1,0,−1,0],
0,0,−31,−31 , 1
2,−21,16,16 ,2
.
Thus in this case, we have two different ideal classes.
Then the class number nis 2, and the left ideals are I1= 12,0,12,0
,
0,12,0,12 ,
0,0,13,13
,[0,0,0,1],1
, I2=
[1,0,1,0],[1,0,−1,0],
0,0,−31,−31 , 1
2,−21,16,16 ,2
.
Now we calculate the right orders: Ri =qrorder(Ii) gives a (maximal) right order for the ideal Ii:
R1=
[1,0,0,0],−1
2 ,0,−12 ,0 ,−1
2 ,−12 ,16,16 , 1
2,−12 ,−16 ,−16 ,1
, R2=
[1,0,0,0],−1
2 ,0,13,−31 ,−1
2 ,0,23,16 , 0,−21,−21,0
,1
.
To calculate the latticesS0i we have to take the trace-zero elements ofZ+ 2Ri. TheZ+ 2Ri are generated by
Z+ 2R1=
[1,0,0,0],[0,0,−1,0],
0,−1,13,13 , 0,−1,−13 ,−13
,1
, Z+ 2R2=
[1,0,0,0],
0,0,23,−13 ,
0,0,43,13 , [0,−1,−1,0],1
. Then we have
S10=
[0,0,−1,0],
0,−1,13,13 ,
0,−1,−13 ,−13 ,1
, S20= 0,0,23,−13
,
0,0,43,13 , [0,−1,−1,0],1
,
and the corresponding theta series gi(τ) = 1
2
b∈Si0
qNb=1 2
x∈Z3
qxtAix,
where Ai is one-half the matrix of the bilinear form Tr(xy) restricted to the lattice Si0. More precisely, if f1, f2, f3is a basis for the latticeL, then the matrixAis given by
A= 1
2Tr(fifj).
With qgram(Si0)/2 we now have the corresponding quadratic forms: We calculateAi= 12qgram(Si0):
A1=
⎡
⎣3 −1 1
−1 23 11
1 11 23
⎤
⎦; A2=
⎡
⎣ 7 −3 −2
−3 11 −4
−2 −4 20
⎤
⎦.
Now we have to calculate the coefficients of the series gi(τ) =1
2
x∈Z3
qxtAix, i= 1,2.
These are computed by the routineqfminim3(Ai, b,0,3), which returns a small vector of lengthb+1 whose (k+1)th component is the number of elements of normk, that is, the k-coefficient of the theta series given by the norm formAi. Then we calculate ten million coefficients of the modular formsgi with 12qfminim3(Ai,10000000,0,3).
3.2 Calculation of the Weight-32 Formg
Once we have these forms, we have to calculate the
“right” linear combination of them. We need the number of units in Ri, which we calculate with qrepnum(Ri,1).
Then we have
w1=qrepnum(R1,1)/2 = 3, w2=qrepnum(R2,1)/2 = 1.
Now we look for the eigenvector of the Brandt matri- ces corresponding to the modular form f defined above:
We know that there exists an eigenvector v of all the Brandt matrices of prime level 17 such thatBp(v) =apv for all primep. The eigenvaluesap are the eigenvalues of the modular formf under the action of the Hecke oper- atorsTp. We use the PARI-GP routineellap(E, p) and calculate
a2=−1, a3= 0, a5=−2, a7= 4, . . . . Then we calculate the Brandt matrixB2:
B2=brandt(R,2) = 0 3
1 2
.
Recall that R is our maximal order in the quaternion algebra ramified at 17 and ∞. Sincea2 =−1, we look for the kernel of
B2−(−matid(2)) = 1 3
1 3
.
This kernel already has dimension one. Then any vector in it, for examplev = (−3,1), will be eigenvector of all Brandt matrices with the required eigenvalues. If it did not have dimension one, we would need to intersect with the kernel of B3−a3I, and so on, until we obtained a dimension-one eigenspace.
Now we have the eigenvectorv= (−3,1) and the val- ues w1 = 3, w2 = 1. Then we set ef = (wv1
1,wv2
2) = (−1,1), and the weight-32 modular form we want is
g=g17= v1
w1g1+ v2
w2g2=−g1+g2. 3.3 |Xd|Formula
We want now to calculate the |X| value of the (−d)- quadratic twists of E. If md is the d-coefficient of the modular formg17, then we know that
L(f,1) Ωf
L(f⊗−d,1)
2Ωf√Im(τ) d
= deg(ϕ)m2d ef, ef ,
and we can calculateef, ef=w1e2f1+w2e2f2= 3(−1)2+ 1·12 = 4. From Cremona’s tables we get deg(ϕ) = 1.
With PARI-GP we calculate L(f,1)/Ωf = 14. So the termL(f,1)/Ωf on the left cancels with deg(ϕ)/ef, ef on the right.
The ratio Ωd/2Ωf√Im(τ)
d does not depend ond, and we have
Ωd= 2ΩfIm(τ)
√d .
The polynomials pd(x) defining the equations of the twisted curvesEdhave negative discriminant. So we have
L(f⊗−d)
2Ωf√Im(τ) d
= L(f⊗−d)
Ωd = |Xd|
pcp,d
|Tor(Ed)|2 . For the order of the group of torsion points of the (−d)- twists ofE, we have|Tor(Ed)|= 2.
Putting all this together gives us
|Xd|= 4m2d
pcp,d,
which gives us a way for calculating, with the coefficients of the formg17, the values of|Xd|. Moreover, since the order ofX is known to be a square,
|Xd|= md
pcp,d
4
gives us a signed square root of|Xd|with the sign given by the coefficient md. This is what we have calculated for different elliptic curves of prime conductors and rank zero.
4. EXPERIMENT AND OBSERVATIONS
By the procedure described above, we can calculate the signed square root of the analytic value |Xd| for the imaginary (−d)-quadratic twistEdof a strong Weil curve E of prime conductorN and rank zero.
This has been done for curves of conductors N = 11,17,19,37,67,73,89,109,139 and for N = 307, this last one in the four existing isogeny classes of curves all of rank zero.
To be more precise, we pick a strong Weil curve E of prime conductorNand rank zero, and calculate between 3 and 10 million coefficients of the weight-32modular form g associated with the weight-2 modular form f of the elliptic curveE.
For thosed such that (−d) is a fundamental discrim- inant and −d
N
= 1 (that is, the sign of the functional equation is +1), we have by the Gross formula and the BSD conjecture that either bothL(Ed,1) andmd equal zero, or both are nonzero and we have a relationship
|Xd| = q2dm2d, where qd is a rational number that in- volves the product of the fudge factors cp,d. Then the (integer) numberqdmd is a signed square root of the or- der of the groupXdwith the sign given by the coefficient md. We will denote this number by Sd:
Sd=
qdmd ifL(Ed,1)= 0, 0 ifL(Ed,1) = 0.
The zero coefficientsmd with (−d) a fundamental dis- criminant and −d
N
= 1 correspond to nontrivial zeros ofL(Ed,1), so by counting these coefficients, we can ob- tain the density of curves Ed such that the L-function vanishes nontrivially ats= 1.
From this information we made graphs for the density distribution of the Sd values obtained (over all of thed’s with (−d) fundamental and−d
N
= 1).
4.1 Observations
Conrey, Keating, Rubinstein, and Snaith [Conrey et al.
04] have obtained conjectures for the value distribu- tion of the Fourier coefficients md based on conjectures from random-matrix theory for the value distribution of L(Ed,1). They observed that for primesp dividing the order of the torsion group of E, the probability that the Fourier coefficientmdis divisible bypdeviates from De- launay’s prediction for the probability of |X| being di- vided by a given prime.
In [Rubinstein 02], Rubinstein makes a graph of the distribution of the coefficients md of the modular form of weight 32, normalized by the product of the fudge fac- tors, for the elliptic curve of conductorN = 11. He points out that in the histogram it is seen that primes 2 and 5 behave differently. He also compares the density of coef- ficients divided by primes p with Delaunay’s prediction [Delaunay 01] for the probability of |X| being divisible bypamong elliptic curves of rank zero.
We will state here the observations we made from our experimental data and explain the density of Sdvalues di- vided bypwhenpis an odd prime that divides|Tor(E)|.
From the graphs, it is clear that there is a symmetry in the behavior of the positive and negative Sd values. The sign of Sd seems to play no role. For this reason we will sometimes restrict our attention to the positive part in further analysis.
In all examples, the density graphs split essentially into two, and this splitting corresponds to the parity of the Sdvalues. We get two density curves: one for odd val- ues of Sdand one for even values. Further, when the base elliptic curve E has nontrivial torsion, these odd/even density curves also seem to “split” or have a “shadow”
(see Figures 1 and 2).
The order of the group of torsion points ofE(Q) seems to affect the behavior of these graphs in the following manner.
1. Curves E with odd nontrivial torsion: For elliptic curves with conductors 11, 19, and 37, the groups of torsion points of E(Q) have, respectively, orders 5,3,3,
71819
0-160 160
FIGURE 1. Density distribution of Sd for N = 11,
|Tor(E)|= 5.
and these are the only cases of nontrivial odd torsion (as for prime conductors, the order of the group of torsion points is one or two—for conductors 64 plus a square—
except for the three cases above and for conductor 17, in which case the torsion is 4). For these three cases, we get two density curves that “split”: those above correspond to odd values of Sd and those below to even values.
Graphically, it is remarked that those values of X divided by the order of the group of torsion points have a larger density. What we mean by this is (from right to left) to choose a point in the density graph of (say) odd values for (say) conductorN = 11 that has a “slightly”
greater density than its near points in the same “odd- value curve.” The next point on the left with the same
“property” will be the fifth, and so on. The same goes forN = 19, 67, and 3.
2. Curves E with trivial torsion: For conductors 67, 109, 139, and 307 the torsion group of the base ellip- tic curvesE is trivial, and we get two well-differentiated density curves, separated into odd values of Sd for the upper density curve and even values of Sdfor the bottom one.
3. Curves E with even torsion: The behavior for con- ductors 17, 73, and 89 is different. The odd/even den- sity curves cross each other. The orders of the groups of torsion points of the elliptic curves are, respectively, 4, 2, and 2. In these cases, small even values of Sd have greater densities than the odd ones; then they are equal, and for larger values, odd Sd’s have greater densities than even ones.
This “crossing” of the even/odd density curves is seen more clearly in the logarithmic graph, as shown in Fig- ure 3. Here again, the odd/even density graph splits, with those Sd divided by |Tor(E)| = 4 or 2, depending on the case, having greater densities than the remaining ones.
All of this raises the following question: are Sd values divided by 5 more frequent forN = 11 than for the other
93471
0-120 120
FIGURE 2. Density distribution of Sd for N = 67,
|Tor(E)|= 1.
conductors calculated? One has the same question for 3 and conductorsN = 19,37, for 4 and conductorN = 17, and for 2 and conductorsN= 73,89.
These questions are the motivation for Table 1, in which we calculated for each conductor the density of those values Sd divided by all the orders of the torsion groups as well as other small primesp.
In the table, T is the order of the group of torsion points, I denotes the isogeny class, and mult n stands for the density of curvesEd whose Sd value is a multiple of n. “Zeros” is the density of curves Ed with positive analytic rank, that is, the density of those curves in which the corresponding L-series vanishes nontrivially at the symmetry centers = 1 (recall that we are looking only at quadratic twists in which the sign of the functional equation is +1), and finally,mdenotes the total number (in millions) ofd’s calculated.
CLis a Cohen–Lenstra heuristic on class numbers and D is Delaunay’s heuristic for |X| being divisible by a primep. We will refer to this in the next sections.
This table is to be read by columns. For example, there is, in general, around 40% of even Sd values, and this percentage increases to 50% or more when the elliptic curveE has even torsion (N = 17,73,89). If we look at Sd values divided by 3, they are in the examples about 35%, except for conductors N = 19,37, in which the group of torsion points has order 3 and this percentage increases to 40%. Something similar is seen for Sd values divided by 4 and 5.
We will now give heuristics for the density of Xval- ues divided by the order of the group of torsion points for conductors 11, 19, and 37 by the Cohen–Lenstra heuris- tics [Cohen and Lenstra 84] on class groups of number fields.
4.2 The Hurwitz Class Number
Let−dbe a negative discriminant andK the imaginary quadratic extension ofQ, K =Q(√
−d). Let O be an
order of discriminant −din K, and denote byh(O) the order of the finite group Pic(O) and by u(O) the order ofO∗/Z∗. Thenu(O) = 1 unlessd=−3,−4, which give 3,2 respectively.
The Hurwitz class number is given by H(d) =H(O) =
O⊂O⊂OK
h(O) u(O).
As stated in [Gross 87], the primeNis used to define the modified invariantHN(d) by
HN(d) =
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
0 ifN splits inO, H(O) ifN is inert inO,
1
2H(O) ifN is ramified inObut does not divide the conductor ofO, HN(d/N2) ifN divides the conductor
ofO,
which is zero unless −d ≡ 0,1 (mod 4) and −d
N
= 1, and ifW =n
i=1wi, thenW HN(d) is integral.
If dis such that (−d) is a fundamental discriminant, thenO=OK andH(OK) =hd/u(OK), whereOK is the ring of integers in K and hd is the class number. Fur- thermore, if we request that−d
N
= 1, thenN is either inert or ramifies inOK and thenHN(d) becomes
HN(d) =
⎧⎨
⎩
hd
u(OK) ifN is inert in K,
1 2 hd
u(OK) ifN is ramified inK.
Recall thatu(OK) = 1, except for exactly 2 values ofd.
4.3 The Cohen–Lenstra Heuristics and |X|Divisibility by 5 for ConductorN =11
LetGbe the weight-32 Eisenstein series defined by G=
n i=1
1 wigi,
where gi are the theta series of the lattices Si0 in the quaternion algebra ramified atN and∞, andwi is half the number of units of the right orders Ri, as stated previously.
Gross has proved thatGhas Fourier expansion G=N−1
24 +
d>0
HN(d)qd
and mentions, forN = 11, the following congruence be- tween the modular form of weight32,g11, and the weight-
3
2 Eisenstein seriesG11:
2g11=−2g1+ 2g2≡3g1+ 2g2= 6G11(mod 5M∗)
N T I zeros mult 2 mult 3 mult 4 mult 5 mult 7 mult 11 m
11 5 A 0.042 0.369 0.353 0.185 0.234 0.146 0.097 10
17 4 A 0.079 0.592 0.353 0.303 0.210 0.156 0.115 10
19 3 A 0.065 0.371 0.407 0.189 0.206 0.150 0.105 3
37 3 B 0.054 0.440 0.413 0.223 0.207 0.149 0.101 10
67 1 A 0.058 0.403 0.355 0.204 0.208 0.150 0.103 10
73 2 A 0.086 0.539 0.356 0.269 0.212 3
89 2 B 0.072 0.507 0.354 0.257 0.209 3
109 1 A 0.076 0.391 0.322 0.199 0.208 0.153 0.109 3
139 1 A 0.073 0.394 0.354 0.192 0.209 3
307 1 A 0.091 0.390 0.354 0.200 0.210 3
307 1 B 0.087 0.405 0.354 0.208 0.210 3
307 1 C 0.076 0.391 0.354 0.199 0.207 3
307 1 D 0.071 0.401 0.354 0.205 0.209 3
D - - - 0.580 0.360 - 0.206 0.145 0.091
CL - - - - 0.439 - 0.239 0.163 0.099
26 3 A 0.043 - 0.415 - 0.207 0.147 0.098 10
26 7 B 0.036 - 0.353 - 0.207 0.160 0.095 10
TABLE 1. Density ofSdvalues divisible by 2,3,4,5,7,11.
(where M∗ is the lattice defined in Section 1), and in particular,
md≡3H11(d) (mod 5).
We shall now use this congruence and the Cohen–
Lenstra heuristics for class numbers to explain the den- sity of|X|divisible by 5 in the family of quadratic twists of the elliptic curve [0,−1,1,−10,−20] of conductor 11.
The order ofXd is given bym2ddivided by a power of 2.
So the density ofSd values divided by 5 is the density of themd’s divided by 5.
Denote byhd the class number of the quadratic field Q(√
−d). Then the congruence above shows that
{d:md ≡0 (mod 5)}={d:hd≡0 (mod 5)}, where both sets are taken overd’s with−da fundamental discriminant and−d
11
= 1. If we make the assumption that class number being divisible by a prime pand hav- ing a particular Kronecker symbol are independent facts, which at least numerically is so, then we have
|
0< d < X:hd≡0 (mod 5) and −d
11
= 1
|
|
0< d < X:−d
11
= 1
|
∼ | {0< d < X:hd≡0 (mod 5)} |
| {0< d < X} |
as X → ∞, where, as before, the sets are taken over d’s such that−dis a fundamental discriminant.
Here is where the Cohen–Lenstra heuristics come in [Cohen and Lenstra 84]. These are heuristics for the probability that the class number of a quadratic imag- inary extension ofQis divisible by a primep:
X→∞lim
| {0< d < X:hd≡0 (modp)} |
| {0< d < X} | =f(p),
where f(p) = 1−
i≥0
1− 1
pi
=1 p+ 1
p2 − 1 p3 − 1
p7 +· · ·. By the assumption made above we can say thatf(5)≈ 0.239 is the probability ofSd being divisible by 5 among negative quadratic twists for conductorN = 11.
4.4 |X|Divisibility by 3 forN =19 andN =37 This exact argument explains the density ofXvalues di- vided by 3 for conductorsN = 19 andN = 37, since the following congruence holds among the respective weight-
3
2 modular form and Eisenstein series, for both conduc- tors
g≡G(mod 3M∗).
Then we have in these two cases that the density ofX values divisible by 3 isf(3)≈0.439.
This is general and can also be applied when the con- ductorN is not prime: ifpis an odd prime that divides
|Tor(E)|, then the density of |X| ≡ 0 (modp) in the family of negative quadratic twists ofE is given by the Cohen–Lenstra heuristics on class numbers in quadratic imaginary extensions ofQ being divisible by p. We are still working out the details and will return to the topic in the future.
In order to compare data, in Table 1 we have cal- culated the densities of |X| divisible by p for curves 26A and 26B (as in Cremona’s tables), which have, re- spectively, groups of torsion points of orders 3 and 7.
The ternary forms and linear combination that gives the weight-32 modular forms have been taken from Tornaria’s data [Tornaria 04]. We restricted for simplicity to d’s coprime to the conductor, and we have an additional re- striction on the sign of
−d p
for primes p dividing N.
Only these can be obtained by this method.
-2.3782
-13.084
0 125
FIGURE 3. Logarithmic graph for curve E of conductor N= 17 of even torsion.
It is clearly seen that p = 3 and p = 7 have bigger density for curves 26A and 26B respectively, and though the size of the experiment is rather small, these densities are not far from Cohen–Lenstra’s prediction. We did not take into accountp= 2, since we have not divided by the fudge factors to obtain the order ofX.
For completeness we also compared our results with Delaunay’s heuristics for the probability of the order of X being divided by a prime p. We remark that De- launay’s heuristics are for elliptic curves of rank zero, ordered by conductor. However, for small odd primes p this prediction seem to be applicable to families of even- rank negative quadratic twists of an elliptic curve. This density is denoted by Din Table 1. Although forp= 2 this prediction does not seem to apply, since in general the density of even values ofXis less than 50%, it does seem to be applicable for N = 17. If we are to suppose that if a primepdivides Tor(E) then the density of|X|
values divided bypis bigger than if it does not, then this case goes in the same direction, since the group of torsion points ofE is of order 4.
4.5 More Observations
Our next question refers to the nature of the density curves. In trying to understand the density distribution and expecting an exponential type, we made graphs for the logarithm of positive Sd values. In the logarithmic graphs one can see clearly the consequences of the pre- vious assertion about the curves corresponding to even and odd values of Sd crossing each other (see Figure 3) for twists of elliptic curvesE of even torsion, instead of the typical situation of odd Sd values remaining above even ones.
Though for the first examples calculated (N = 11,17,19,37), these logarithmic density graphs seemed to behave linearly for even values of Sd, for N = 67 this was not the case. In other examples (N = 73,89,109,139,307), it is seen that we cannot assume a linear behavior, since for small values of Sd we get a
-2.8033
-14.249
0 261
FIGURE 4. Logarithmic graph for conductorN = 67.
-3.1406
-16.134
0 239
FIGURE 5. Logarithmic graph and best linear approxima- tion for conductorN= 11.
(greater or lesser) separation (above) from the line. In- stead, it is clear that the logarithm of odd Sd values is not linear, having greater densities for small values and aligning with the even-Sd graph. There is one exception to this, for conductor N = 37, in which both even and odd logarithmic values behave almost linearly. We ob- serve, for what it’s worth, that this elliptic curve is the only one in the examples analyzed that has two connected components over the real numbers.
The typical situation for curves E of trivial or odd torsion is shown in Figures 4 and 5. The situation is the same for density logarithmic graphs of curvesE of even torsion.
4.6 About Theta Series
Graphs have been made for densities of the coefficients ad of each of the weight-32 theta seriesgi involved in the linear combination that gives the theta series g associ- ated with the elliptic curve E. Most of the coefficients are zero. Between the nonzero ones there is no evidence of the order of the group of torsion points of the ellip- tic curve E in the density of the coefficients divided by 2,3,4,5 as in the S-values, even if we restrict to thosed’s that are fundamental discriminants and with the correct Kronecker symbol.
It is surprising, however, that for conductor 11 the density distribution for the coefficients of one of the theta series is a graph that separates into five density curves, which is the order of the group of torsion points. Odd
11394
00 6804
FIGURE 6. Density distribution of theta coefficients g1; ternary form has four automorphisms,Eof conductor 11.
7529
01 2124
FIGURE 7. Density distribution of theta coefficients g1; ternary form has four automorphisms,Eof conductor 19.
coefficients are insignificant, and the even ones split ac- cording to their congruence modulo 24. The same occurs forN = 19 and 37: the graph for the density distribution of the coefficients of the theta series splits into three, cor- responding to the congruence of the coefficients modulo 8 or 4 (depending on the case). ForN = 17 we also have a graph that splits into five density curves, similar to that for conductor 11, according to congruency modulo 24.
The only difference we can point out at the moment in the examples calculated is that when the order of the group of torsion points ofE is greater than 2, we obtain
“cloudy” graphs.
When the elliptic curve E has torsion 1 or 2, the sit- uation in the examples calculated seems to be clearer.
We get neat graphs and the shapes of these density co- efficient graphs seem to depend on the number of au- tomorphisms of the corresponding ternary form. The density graphs for nonzero coefficients divide essentially in two, depending on the number of automorphisms of the ternary form involved. For ternary forms that have one automorphism, the coefficients split essentially into even/odd ones, with major density for the even ones. For two automorphisms, odd coefficients are in a much lower proportion than even ones. The density curve for the even coefficients splits in two, depending on congruence modulo 4. For four automorphisms, odd coefficients are in an insignificant proportion. The typical situation is ex- emplified in Figures 8 and 9 for a curveE of conductor
18958
01 1154
FIGURE 8. N = 67, coefficient density for theta series of ternary form with two automorphisms.
12047
01 1168
FIGURE 9. N = 67, coefficient density for theta series of ternary form with one automorphism.
N = 67 of torsion 1. All these graphs can be found on- line at http://www.expmath.org/expmath/volumes/15/
15.3/quattrini/.
Of the graphs available at the web site, twistNrtshadist.ps gives the density distribu- tion of the signed square roots ofXfor the elliptic curve of conductorN with sign given by the d-coefficientmd. The graphs for the density distribution of the coefficients of the theta series gi are in twistNdistgi.ps. The theta series are numbered accordingly with ternary forms, which we do not give explicitly here, but they are at the web site. The graph twistNdistg.psgives the coefficient distribution of the linear combinationg that corresponds tof.
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Patricia L. Quattrini, Universidad de Buenos Aires, Departamento de Matematica, FCEyN, Pab. I, Ciudad Universitaria, 1428 Buenos Aires, Argentina ([email protected])
Received March 3, 2005; accepted October 6, 2005.
